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Product-Type Probability Bounds of Higher Order

Published online by Cambridge University Press:  27 July 2009

Hentry W. Block ,
Tim Costigan  and
Allan R. Sampson
Hentry W. Block
Affiliation:
Department of Mathematics and StaatisticsUniversity of Pittsburgh Pittsburgh, Pennsylvania 15260
Tim Costigan
Affiliation:
Department of StatisticsThe Ohio State University, 141 Cockins Hall 1958 Neil Avenue Columbus, Ohio 43210-1247
Allan R. Sampson
Affiliation:
Department of Mathematics and StatisticsUniversity of Pittsburgh Pittsburgh, Pennsylvania 15260
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Abstract

GIaz and Johnson [14] introduce ith-order product-type approximations, βi, i =1,…n − 1, for Pn = P(X1 ≤c1, X2 ≤ c2,…Xn≤ cn) and show that Pn≥ βn−1 ≥ βn−2 ≥… ≥ β2 ≥ β1 when X is MTP2. In this article, it is shown that

under weaker positive dependence conditions. For multivariate normal distributions, these conditions reduce to cov(Xi,Xj) ≥ 0 for 1 ≤ i < j ≤ n and cov(Xi,Xj| Xj−1) ≥ 0 for 1 ≤ i < j − 1, j = 3,…,n. This is applied to group sequential analysis with bivariate normal responses. Conditions for Pn ≥ β3 ≥ β2 ≥β1 are also derived. Bound conditions are also obtained that ensure that product-type approximations are nested lower bounds to upper orthant probabilities P(X1 > C1,…,Xn > cn). It is shown that these conditions are satisfied for the multivariate exponential distribution of Marshall and Olkin [20].

Information

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 6 , Issue 3 , July 1992 , pp. 349 - 370
Copyright
Copyright © Cambridge University Press 1992

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References

Anderson, T.W. (1984). An introduction to multivariate statistical analysis. New York: Wiley.Google Scholar
Barlow, R.E. & Proschan, F. (1981). Statistical theory of reliability and life testing probability models. Silver Springs, MD: To Begin With.Google Scholar
Bauer, P. (1986). Two stage testing for simultaneously testing main and side effects in clinical trials. Biometrical Journal 28: 871–879.CrossRefGoogle Scholar
Bauer, P. & Hackl, P. (1985). The application of Hunter's inequality in simultaneous testing. Biometrical Journal 27: 25–38.CrossRefGoogle Scholar
Block, H.W. & Sampson, A.R. (1988). Conditionally ordered distributions. Journal of Multivan ate Analysis 27: 91–104.CrossRefGoogle Scholar
Block, H.W. & Ting, M.L. (1981). Some concepts of multivariate dependence. Communtcations in Statistics, Series A 10: 749–762.Google Scholar
Bolviken, E. (1982). Probability inequalities for the multivariate normal with non-negative partial correlations. Scandinavian Journal of Statistics 9: 49–58.Google Scholar
Chhetry, D. (1983). Some concepts of setwise positive dependence. Ph.D. dissertation, University of Texas at Dallas.Google Scholar
Chhetry, D., Kimeldorf, G., & Sampson, A.R. (1989). Concepts of setwise dependence. Probability in the Engineering and Informational Sciences 3: 367–380.CrossRefGoogle Scholar
Cornish, E.A. (1954). The multivariate t-distribution associated with a set of normal deviates. Australian Journal of Physics 7: 531–542.CrossRefGoogle Scholar
Coostigan, T.M. (1988). Product-type probability bounds. Ph.D. dissertation, University of Pittsburgh.Google Scholar
Dunnett, C.W. & Sobel, M. (1954). A bivariate generalization of student's t-distribution, with tables for certain special cases. Biometrika 41: 153–169.CrossRefGoogle Scholar
Esary, J.D., Proschan, F., & Walkup, D. (1967). Association of random variables with applications. Annals of Mathematical Statistics 38: 1466–1474.CrossRefGoogle Scholar
Glaz, J. & Johnson, Mck, B.. (1984). Probability inequalities for multivariate distributions with dependence structures. Journal of the American Statistical Association 79: 436–441.CrossRefGoogle Scholar
Glaz, J. & Johnson, B. McK. (1986). Approximating boundary crossing probabilities with application.to sequential tests. SequenlialAnalysis 5 (No. 1): 37–72.Google Scholar
Karlin, S. & Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities- 1: Multivariate totally positive distributions. Journal of Multivariate Analysis 10: 476–498.Google Scholar
Karlin, S. & Rinott, Y. (1983). M-matrices as covariance matrices of multinormal distributions. Linear Algebra and Its Applications 52/53: 419–438.CrossRefGoogle Scholar
Kenyon, J.R. (1987). Bounds and approximations for multivariate probabilities: Their calculation and applications. Ph.D. dissertation, University of Connecticut, Storrs.Google Scholar
Lee, M. L. Ting (1985). Dependence by total positivity. Annals of Probability 13: 572–582.CrossRefGoogle Scholar
Marshall, A. & Olkin, I. (1967). A multivariate exponential distribution. Journal of the American Statistical Association 62: 30–44.CrossRefGoogle Scholar
Milhako, D. (1987). Repeated significance testing in survival analysis using litter-matched design. Biometrics 43: 115–125.CrossRefGoogle Scholar
Nauss, J. I. (1982). Approximations for distributions of scan statistics. Journal of the American Statistical Association 77: 177–183.CrossRefGoogle Scholar
Sampson, A.R. (1983). Positive dependence properties of elliptically symmetric distributions. Journal of Multivariate Analysis 13: 375–381.CrossRefGoogle Scholar
Sarkar, S.K. & Smith, W. (1986). Probability inequalities for ordered MTP2 random variables. Sankya 48 (Ser. A. Pt. 2): 119–135.Google Scholar
Sidak, Z. (1967). Rectangular confidence regions for means of multivariate normal distributions. Journal of the American Statistical Association 62: 626–633.Google Scholar
Slepian, D. (1962). On one-sided barrier problem for Gaussian noise. Bell System Technical Journal 41: 463–501.CrossRefGoogle Scholar
Thall, P.F., Simon, R., & Ellenberg, S.S. (1988). Designs for comparative clinical trials. Biometrika 75: 303–310.CrossRefGoogle Scholar
Tong, Y.L. (1980). Probability inequalities in multivariate distributions. New York: Academic Press.Google Scholar